© 2000 John Petroff
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© 2000 John Petroff |
Rates of return play a key role in value determination. In many cases they are the relative measures of value themself. There are many different definitions of rate of return. The following is a review of these definitions. Most should be familiar to anyone exposed to finance. The comprehensive listing is presented here so that no ambiguity exists when rates of return are mentioned throughout the following chapters. Only major variations are covered without close attention to certain modifications in calculation (such as those pertaining to the timing of payments, e.g. annually, semi-annually or monthly) which are dealt with elsewhere in this text.
A first distinction can be made between market rates and rates calculated on the basis of elements specific to a given financial asset. A second distinction is between rates that are future oriented, and those that are based on historical data. It has already been extensively argued in Section A and in Section B-2 of this chapter, in particular, that future returns are what matters, and past returns are irrelevant. But future returns can only be estimated with the help of what has actually been received in the past, or what has been promised. We start with the rates that are promised when a financial asset is created.
These include all forms of contractual promises to pay. Except for variable rates and participating preferred stock dividends, these rates is not affected by future changes in company or market conditions.
a) - Effiective rates on deposit accounts, saving accounts, loans, mortgages and other financial contracts
In addition to the nominal interest rate, an effective annual rate or APR (annual percentage rate) must be shown in all contracts of American financial intermediaries. The effective annual rate is different from the nominal rate whenever interest accrues more often than once a year.
APR = (1 + i/m)m -1
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A mortgage loan stipulates a rate of 9% with interest charged monthly. The effective annual rate is APR = (1 + 0.09/12)12 - 1 = 1.09381 - 1 = 0.09381 or 9.38% |
b) - Discount on money market instruments
This definition of discount rate, which is sometimes called quoted yield, applies especially for government securities and commercial paper, and most securities with maturity less than one year. In the United States, the discount rate d is stated on a 360 days year basis in all discounted financial instruments. It may often be necessary to calculate the actual interest rate on a 365 days year basis, which is called simple interest rate.The dollar discount DD is calculated with the exact number of days t remaining to maturity.
DD = Pr . d (t/360)
The simple interest rate SI is equal to
SI = (DD / (Pr-DD)) (365 / t)
or replacing DD
SI = (Pr . d(t/360))/ (Pr - Pr . d (t/360))) (365 / t) which simplifies to
SI = 365 d / (360 - t d)
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A 180 days treasury bill of $10.000 face value is sold at a discount rate of 6.80%. The dollar discount is DD = 10,000 x 0.068 (180 /360) = 10,000 x 0.034 = $340.00 The purchase price is P = 10,000 - 340 = $9,660.00 The simple interest is SI = 365 x 0.068 / (360 - 180 x 0.068 ) = 24.82 / 347.76 = 0.07137 or 7.14% One can verify the calculation of the simple interest rate by deriving the implied interest rate in the purchase price of $9,660 for 180 days = ((10,000 - 9,660)/9,660)(365/180) = (340/9,660)(365/180) = 0.0351967 x 2.0277778 = 0.07137 |
Note: this discount rate has nothing to do with the rate charged by Federal Reserve Banks, which is also known as discount rate and which is essentially a tool of monetary policy.
The coupon rate CR is calculated by dividing the sum of coupon payments in one year C by principal or face value of the bond Pr. The coupon rate remains constant throughout the life of the bond.
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In 1995, the coupon rate of ATT 6 3/4, 2005 is 6 3/4. It determines the amount of the coupon paid, which is $675 on a $10,000 face value bond. Or, if we are given the annual coupon of $675 CR = 675 / 10,000 = 0.0675 or 6.75% Since coupon and principal are unchanging the coupon rate is the same in any other year as well. |
d)- Guaranteed preferred stock dividends
Preferred stock are indeed preferred because their dividends are guaranteed, which is never the case for common stock. The guarantee is expressed in a promised dollar amount to be paid on specified dates regardless of corporate annual results. Sometime, instead of the dollar amount, the promise is stated as a rate of a stated par value, and it is similar to the coupon rate. Most preferred stock dividends are also promised to be cumulative which means that missed dividends must be paid in subsequent years and before any common stock dividends is distributed.
e)- Participating preferred stock
Some preferred stock dividend is stated to be participating. This means that, in addition to the fixed guaranteed amount, preferred shareholders also receive a portion of profits, if any, together with or after common shareholders. The provision is relatively rare and the formula for calculating the additional dividend vary considerably.
f)- Indexed and variable rates
When inflation picks up, many debt instruments (especially mortgages and credit cards) are issued with variable rates to protect borrowers and especially lenders (because variable rates protect banks from interest rate risk). After inflation subsides, lenders offer a choice of variable or fixed rate contracts. The rate is not entirely variable, but is a combination of a fixed portion and variable portion. The variable portion is usually stated as a percentage of a major market rate (or index) averaged over a period of six months or a year.
VR = f + p(I)
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For example, a mortgage provides for a variable rate calculated as % plus 50% of average TB rates over past six months. Many credit cards offer variable rates calculated as 10.5% plus prime rate published in the press. |
A common index is the Treasury Bills rate in the United States, and similar government securities in other countries. The adjustment in rate can be annual, monthly or continuous. In the United States decreases were immediate but increases delayed until the end of the month when instituted in the 1980's. Some loans have floor and ceiling clauses to prevent spikes in rates and protect the parties.
Most borrowers prefer fixed rates which are consequently always slightly higher than variable rates for same contracts. Borrower prefer fixed rates because they protect from future inflation, and most contracts can be refinanced should rates happen to dip lower.
Here are included rates of return resulting from actual payments received over a period of time. These rates are based on past statistics and as such do not change unless the period covered is expanded with more recent numbers.
a)- Rate of return on stock ownership
The rate of return is the combination of price change and distribution received:
Rs = (cash received + (Ending price - Beginning price)) / Beginning price
Although rates of return reported in the press are sometimes calculated this way, one would immediately notice that this definition is imperfect because it does not account for the length of time the financial asset was held, and because it does not pay attention to time value of money.
b)- Holding period return
Holding period return was presented in Section A-2 of this chapter. It is a direct application of the previous definition with an assumption that the holding period is equal to or less than one year (i.e. discounting for loss of purchasing power is not necessary).
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IBM stock is purchased on March 3, 1994 for $150.00 and sold on November 16, 1994 for $156.00 after receiving a dividend of $3.25. The holding period return is HRP = (3.25 + (156.00 - 150.00 ) ) / 150 = (3.25 + 6.00) / 150 = 9.25 / 150 = 0.06167 or 6.2% |
c)- Annual rate of return equivalent of holding period return
Rt = Rh(365 / holding period)
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Continuing with IBM stock purchased on March 3, 1994 for $150.00 and sold on November 16, 1994 for $156.00 after receiving a dividend of $3.25. The annual rate of return over the holding period of 257 days is Rt = 0.06167 (365 / 257) = 0.06167 x 1.4202 = 0.08758 or 8.8% |
d)- Arithmetic mean annual rate of return
Ra = Sum(Rt) / n
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For example, XYZ stock was purchased in 1995 at $80 and sold in 2001 for $115 after receiving $6 of dividends each year for the six years. The total holding period return is Rh = (36 + (115 - 80)) / 80 = 71 / 80 = .8875 The arithmetic mean rate of return is Ra = .8875 / 6 = .1479 or 14.79% |
e)- Geometric mean annual rate of return
Rg = ((Pt +Sum(cash received))/ P0)1/n -1
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In the previous example the geomeric mean annual return is Rg = ((115 + 36) / 80)1/3 - 1 = (151 / 80)1/3 - 1 = (1.8875)1/3 - 1 = 1.11168 - 1 = .11168 or 11.17% |
3)- Current rates
These are rates that pay special attention to the current market price of financial assets. As such, these rates go up and down with every price change of the security in the market.
The current yield is calculated by dividing the sum of coupon payments received or promised in the current year by the latest quoted price of the bond.
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Let us return to the example used in Section C-2 of this chapter. In 1995, the current yield of ATT 6 3/4, 2005 is CY = 675 / 9,824 = .06871 or 6.9 % |
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We continue with ATT 6 3/4, 2005 priced at $9,824 in 1995, but we assume that the coupon is paid twice a year. The amount paid each six months is $337.50 (or half of $675). The rolling yield RY is the sum of - current yield which was previously calculated as CY = 0.06871 - reinvested coupon of $337.50 for six months at prevailing market rate of 7% earns Rri = (337.50 x 0.07) / 9,824 = 0.00240 - appreciation of bond over one year assuming market rates remain unchanged; at the end of the year the bond with nine years remaining and semi-annual coupon of 337.50 would sell for P1 = 337.50 (1 - 1/(1 + 0.035)18 )/0.035 + 10000 / (1 + 0.035)18 = 9,835.13 and the appreciation over one year is RP1-P0.= (9,835 - 9,824) / 9,824 = 0.00113 Thus, the rolling yield is RY = 0.06871 + 0.00240 + 0.00113 = 0.07224 or 7.2% |
Dividend yield is calculated by dividing the dividend for the current year (either the last dividend received or the dividend declared by the corporation) by the most recent traded (i.e. closing price at the end of latest market session) price of the stock.
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Dividend yield of IBM stock priced at $156.00 on November 16, 1994 and paying an annual dividend of $3.25 is Rd = 3.25 / 156 = 0.02083 or 2.1% |
4)- Total return rate of return
This is a future oriented rate which is predicated on the assumption that all promised or projected payments will be carried out and the initial value of the security is the latest market price. This rate of return is affected by changes in both projected payments and market price.
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Let us return to the example used in Section C-2 of this chapter, where a ATT 6 3/4, 2005 is priced at $9,824 in 1995, or exactly ten years prior to its maturity. What is its yield to maturity? Since 9,824 = 675 x (1 - 1/(1 + 0.07)10 )/0.07 + 10000
/ (1 + 0.07)10 For less simplistic cases, the easiest method of calculation is to obtain an internal rate of return in a spreadsheet. |
See definition and calculation of internal rate of return in Chapter 3 Section G-2 and further examples in Chapter 10 Section E-1.
These rates combine company and market considerations.
a)- Stock required rate of return
This rate was defined in Section E-1 of this chapter:
Rsk= RFR + BETA*(Rm-RFR)
b)- Bond industry yields
To make decisions on bond purchases, yields on bonds in the same industry must be obtained. These are avaiable from specialized services or can be found in the press (see Chapter 3 Section A-1).
c)- Weighted average cost of capital
The weighted average cost of capital is commonly used in calculation of company value (see Chapter 3 Section F-1) and capital budgeting (see Chapter 3 Section G-3), and it is presented in Chapter 11 Section D-1.
The weighted average cost of capital WACC is given by
WACC = d * kd (1-T) + e * ke
where d = proportion of debt in total assets
e = proportion of equity in total assets (note that d + e = 1)
kd = long term bond yield
ke = rate of return on common stock
T = average corporate income tax rate
d)- Risk adjusted weighted average cost of capital
This risk adjusted weighter average cost of capital is assigned as discount rate to projects in capital budgeting when projects deal with new products that are different from current company products. The BETA used should be the one of the industry in which the new product will compete.
See Chapter 10 Section E-2 for justification and Chapter 10 Section E-1 for applications.
e)- Marginal cost of capital
The marginal cost of capital is the rate that must be paid on additional funds sought. It is only occasionally used as mentioned in Chapter 3 Section G-3 and Chapter 11 Section D-1.
See review questions Q-2F.1 through Q-2F.8.
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